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   <h4 class="subsectionHead"><span class="titlemark">2.6   </span> <a 
 id="x11-240002.6"></a>Plate &amp; Shell Elements</h4>
<!--l. 686--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.6.1   </span> <a 
 id="x11-250002.6.1"></a>DKT Element</h5>
<!--l. 688--><p class="noindent" >Implementation of Discrete Kirchhoff Triangle (DKT) plate element. This element is suitable for thin plates, as the
traswerse shear strain energy is neglected. The structure should be defined in x,y plane, nodes should be
numbered anti-clockwise (positive rotation around z-axis). The element features are summarized in
Table&#x00A0;<a 
href="#x11-2500118">18<!--tex4ht:ref: dktplatesummary --></a>.
<div class="center" 
>
<!--l. 694--><p class="noindent" >
<div class="table">
                                                                                           
                                                                                           
<!--l. 694--><p class="noindent" ><a 
 id="x11-2500118"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-19" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-19-1g"><col 
id="TBL-19-1"></colgroup><colgroup id="TBL-19-2g"><col 
id="TBL-19-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-19-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-19-1-2"  
class="td11"><!--l. 694--><p class="noindent" ><span 
class="cmbx-10">dktplate</span>                                                        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-19-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-19-2-2"  
class="td11"><!--l. 694--><p class="noindent" >2D Discrete Kirchhoff Triangular plate element               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-19-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-19-3-2"  
class="td11"><!--l. 694--><p class="noindent" >-                                                                            </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-19-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-19-4-2"  
class="td11"><!--l. 696--><p class="noindent" >Three  dofs  (w-displacement,  u  and  v  -  rotations)  are
required in each node.                                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-19-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-19-5-2"  
class="td11"><!--l. 697--><p class="noindent" >Quadratic approximation of rotations, cubic approximation
of displacement along the edges. Note: there is no need to
define interpolation for displacement on the element.        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-19-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-19-6-2"  
class="td11"><!--l. 698--><p class="noindent" >Default integration of all terms using three point formula. </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-19-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-7-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-19-7-2"  
class="td11"><!--l. 699--><p class="noindent" >Layered cross section support.                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-19-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-8-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-19-8-2"  
class="td11"><!--l. 700--><p class="noindent" >Cross section thickness is required.                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-19-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-9-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-19-9-2"  
class="td11"><!--l. 701--><p class="noindent" >Body loads are supported. Boundary load support is beta.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-19-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-10-1"  
class="td11">Output                </td><td  style="white-space:wrap; text-align:left;" id="TBL-19-10-2"  
class="td11"><!--l. 708--><p class="noindent" >On output, the generalized shell strain/force momentum
vectors in global coordinate system are printed, with the
following meaning:
<table 
class="align-star">
          <tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">&#x03B5;</span></sub></td>          <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>               <td 
class="align-label"></td>        <td 
class="align-label">
        </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">&sigma;</span></sub></td>          <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmsy-10">}</span></td>    <td 
class="align-label"></td>    <td 
class="align-label"></td></tr></table>
where                                                         <span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">xy</span></sub>
are membrane in plane normal deformations, <span 
class="cmmi-10">&gamma;</span><sub><span 
class="cmmi-7">zx</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xz</span></sub> are
(out of plane and in plane) shear componets, <span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub> are
curvatures, <span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">yz</span></sub> are integral forces (normal
and shear forces), and <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub> are bending moments.
Please  note,  for  example,  that  bending  moment  <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub>  is
defined as <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> = <span 
class="cmex-10">&int;</span>
  <span 
class="cmmi-10">&sigma;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">z</span><span 
class="cmmi-10">&#x00A0;dz</span>, so it acts along the y-axis and
positive value causes tension in bottom layer.
</td></tr><tr  
 style="vertical-align:baseline;" id="TBL-19-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-11-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-19-11-2"  
class="td11"><!--l. 709--><p class="noindent" >0.                                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-19-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-19-12-2"  
class="td11"><!--l. 710--><p class="noindent" >Reliable                                                                  </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-19-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-13-1"  
class="td11">Reference </td> <td  style="white-space:wrap; text-align:left;" id="TBL-19-13-2"  
class="td11"><!--l. 711--><p class="noindent" >J.L.Batoz,
K.J.Bathe, L.W.Ho: A study of three-node triangular plate
bending elements, IJNME, 15(12):1771-1812, 1980           </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-19-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-14-1"  
class="td11">                </td></tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-19-15-"><td  style="white-space:nowrap; text-align:left;" id="TBL-19-15-1"  
class="td11"></td></tr></table>
</div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;18: </span><span  
class="content">DKTplate element summary</span></div><!--tex4ht:label?: x11-2500118 -->
</div><hr class="endfloat" />
</div>
</div>
<!--l. 715--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.6.2   </span> <a 
 id="x11-260002.6.2"></a>QDKT Element</h5>
<!--l. 717--><p class="noindent" >Implementation of Discrete Kirchhoff Theory plate quad element (QDKT). This element is suitable for thin plates, as
the traswerse shear strain energy is neglected. The structure should be defined in x,y plane, nodes should be
                                                                                           
                                                                                           
numbered anti-clockwise (positive rotation around z-axis). The element features are summarized in
Table&#x00A0;<a 
href="#x11-2600119">19<!--tex4ht:ref: qdktplatesummary --></a>.
<div class="center" 
>
<!--l. 723--><p class="noindent" >
<div class="table">
<!--l. 723--><p class="noindent" ><a 
 id="x11-2600119"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-20" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-20-1g"><col 
id="TBL-20-1"></colgroup><colgroup id="TBL-20-2g"><col 
id="TBL-20-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-20-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-20-1-2"  
class="td11"><!--l. 723--><p class="noindent" ><span 
class="cmbx-10">qdktplate</span>                                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-20-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-20-2-2"  
class="td11"><!--l. 723--><p class="noindent" >2D Discrete Kirchhoff Quad plate element                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-20-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-20-3-2"  
class="td11"><!--l. 723--><p class="noindent" >-                                                                            </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-20-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-20-4-2"  
class="td11"><!--l. 725--><p class="noindent" >Three  dofs  (w-displacement,  u  and  v  -  rotations)  are
required in each node.                                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-20-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-20-5-2"  
class="td11"><!--l. 726--><p class="noindent" >Quadratic approximation of rotations, cubic approximation
of displacement along the edges. Note: there is no need to
define interpolation for displacement on the element.        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-20-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-20-6-2"  
class="td11"><!--l. 727--><p class="noindent" >Default integration of all bending terms using four point
formula.                                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-20-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-7-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-20-7-2"  
class="td11"><!--l. 728--><p class="noindent" >Layered cross section support.                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-20-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-8-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-20-8-2"  
class="td11"><!--l. 729--><p class="noindent" >Cross section thickness is required.                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-20-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-9-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-20-9-2"  
class="td11"><!--l. 730--><p class="noindent" >Body loads are supported.                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-20-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-10-1"  
class="td11">Output                </td><td  style="white-space:wrap; text-align:left;" id="TBL-20-10-2"  
class="td11"><!--l. 737--><p class="noindent" >On output, the generalized shell strain/force momentum
vectors in global coordinate system are printed, with the
following meaning:
<table 
class="align-star">
          <tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">&#x03B5;</span></sub></td>          <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>               <td 
class="align-label"></td>        <td 
class="align-label">
        </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">&sigma;</span></sub></td>          <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmsy-10">}</span></td>    <td 
class="align-label"></td>    <td 
class="align-label"></td></tr></table>
where                                                         <span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">xy</span></sub>
are membrane in plane normal deformations, <span 
class="cmmi-10">&gamma;</span><sub><span 
class="cmmi-7">zx</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xz</span></sub> are
(out of plane and in plane) shear componets, <span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub> are
curvatures, <span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">yz</span></sub> are integral forces (normal
and shear forces), and <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub> are bending moments.
Please  note,  for  example,  that  bending  moment  <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub>  is
defined as <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> = <span 
class="cmex-10">&int;</span>
  <span 
class="cmmi-10">&sigma;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">z</span><span 
class="cmmi-10">&#x00A0;dz</span>, so it acts along the y-axis and
positive value causes tension in bottom layer.
</td></tr><tr  
 style="vertical-align:baseline;" id="TBL-20-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-11-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-20-11-2"  
class="td11"><!--l. 738--><p class="noindent" >0.                                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-20-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-20-12-2"  
class="td11"><!--l. 739--><p class="noindent" >Reliable                                                                  </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-20-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-13-1"  
class="td11">Reference </td> <td  style="white-space:wrap; text-align:left;" id="TBL-20-13-2"  
class="td11"><!--l. 740--><p class="noindent" >J.L.Batoz,
K.J.Bathe, L.W.Ho: A study of three-node triangular plate
bending elements, IJNME, 15(12):1771-1812, 1980           </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-20-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-14-1"  
class="td11">                </td></tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-20-15-"><td  style="white-space:nowrap; text-align:left;" id="TBL-20-15-1"  
class="td11"></td></tr></table>
</div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;19: </span><span  
class="content">QDKTplate element summary</span></div><!--tex4ht:label?: x11-2600119 -->
</div><hr class="endfloat" />
                                                                                           
                                                                                           
</div>
</div>
<!--l. 743--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.6.3   </span> <a 
 id="x11-270002.6.3"></a>CCT Element</h5>
<!--l. 745--><p class="noindent" >Implementation of constant curvature triangular element for plate analysis. Formulation based on Mindlin hypothesis.
The structure should be defined in x,y plane. The nodes should be numbered anti-clockwise (positive rotation around
z-axis). The element features are summarized in Table&#x00A0;<a 
href="#x11-2700120">20<!--tex4ht:ref: cctplatesummary --></a>.
<div class="center" 
>
<!--l. 751--><p class="noindent" >
<div class="table">
                                                                                           
                                                                                           
<!--l. 751--><p class="noindent" ><a 
 id="x11-2700120"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-21" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-21-1g"><col 
id="TBL-21-1"></colgroup><colgroup id="TBL-21-2g"><col 
id="TBL-21-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-21-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-21-1-2"  
class="td11"><!--l. 751--><p class="noindent" ><span 
class="cmbx-10">cctplate</span>                                                        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-21-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-21-2-2"  
class="td11"><!--l. 751--><p class="noindent" >2D constant curvature triangular plate element               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-21-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-21-3-2"  
class="td11"><!--l. 751--><p class="noindent" >-                                                                            </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-21-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-21-4-2"  
class="td11"><!--l. 753--><p class="noindent" >Three  dofs  (w-displacement,  u  and  v  -  rotations)  are
required in each node.                                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-21-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-21-5-2"  
class="td11"><!--l. 754--><p class="noindent" >Linear      approximation      of      rotations,      quadratic
approximation of displacement.                                    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-21-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-21-6-2"  
class="td11"><!--l. 755--><p class="noindent" >Integration of all terms using one point formula.             </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-21-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-7-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-21-7-2"  
class="td11"><!--l. 756--><p class="noindent" >Layered cross section support.                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-21-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-8-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-21-8-2"  
class="td11"><!--l. 757--><p class="noindent" >Cross section thickness is required.                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-21-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-9-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-21-9-2"  
class="td11"><!--l. 758--><p class="noindent" >Body  loads  are  supported.  Boundary  loads  are  not
supported now.                                                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-21-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-10-1"  
class="td11">Output                </td><td  style="white-space:wrap; text-align:left;" id="TBL-21-10-2"  
class="td11"><!--l. 765--><p class="noindent" >On output, the generalized shell strain/force momentum
vectors in global coordinate system are printed, with the
following meaning:
<table 
class="align-star">
          <tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">&#x03B5;</span></sub></td>          <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>               <td 
class="align-label"></td>        <td 
class="align-label">
        </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">&sigma;</span></sub></td>          <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmsy-10">}</span></td>    <td 
class="align-label"></td>    <td 
class="align-label"></td></tr></table>
where                                                         <span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">xy</span></sub>
are membrane in plane normal deformations, <span 
class="cmmi-10">&gamma;</span><sub><span 
class="cmmi-7">zx</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xz</span></sub> are
(out of plane and in plane) shear componets, <span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub> are
curvatures, <span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">yz</span></sub> are integral forces (normal
and shear forces), and <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub> are bending moments.
Please  note,  for  example,  that  bending  moment  <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub>  is
defined as <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> = <span 
class="cmex-10">&int;</span>
  <span 
class="cmmi-10">&sigma;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">z</span><span 
class="cmmi-10">&#x00A0;dz</span>, so it acts along the y-axis and
positive value causes tension in bottom layer.
</td></tr><tr  
 style="vertical-align:baseline;" id="TBL-21-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-11-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-21-11-2"  
class="td11"><!--l. 766--><p class="noindent" >0.                                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-21-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-21-12-2"  
class="td11"><!--l. 767--><p class="noindent" >Reliable                                                                  </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-21-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-13-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-21-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-21-14-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;20: </span><span  
class="content">cctplate element summary</span></div><!--tex4ht:label?: x11-2700120 -->
</div><hr class="endfloat" />
</div>
</div>
<!--l. 771--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.6.4   </span> <a 
 id="x11-280002.6.4"></a>CCT3D Element</h5>
<!--l. 772--><p class="noindent" >Implementation of constant curvature triangular element for plate analysis. Formulation based on Mindlin hypothesis.
The element could be arbitrarily oriented in space. The nodes should be numbered anti-clockwise (positive rotation
around element normal). The element features are summarized in Table&#x00A0;<a 
href="#x11-2800121">21<!--tex4ht:ref: cctplate3dsummary --></a>.
                                                                                           
                                                                                           
<div class="center" 
>
<!--l. 777--><p class="noindent" >
<div class="table">
<!--l. 777--><p class="noindent" ><a 
 id="x11-2800121"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-22" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-22-1g"><col 
id="TBL-22-1"></colgroup><colgroup id="TBL-22-2g"><col 
id="TBL-22-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-22-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-22-1-2"  
class="td11"><!--l. 777--><p class="noindent" ><span 
class="cmbx-10">cctplate3d</span>                                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-22-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-22-2-2"  
class="td11"><!--l. 777--><p class="noindent" >Constant  curvature  triangular  plate  element  in  arbitray
position                                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-22-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-22-3-2"  
class="td11"><!--l. 777--><p class="noindent" >-                                                                            </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-22-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-22-4-2"  
class="td11"><!--l. 779--><p class="noindent" >Six dofs (u,v,w-displacements and u,v,w rotations) are in
general required in each node.                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-22-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-22-5-2"  
class="td11"><!--l. 780--><p class="noindent" >Linear      approximation      of      ratations,      quadratic
approximation of displacement.                                    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-22-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-22-6-2"  
class="td11"><!--l. 781--><p class="noindent" >Integration of all terms using one point formula.             </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-22-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-7-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-22-7-2"  
class="td11"><!--l. 782--><p class="noindent" >Layered cross section support.                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-22-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-8-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-22-8-2"  
class="td11"><!--l. 783--><p class="noindent" >Cross section thickness is required.                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-22-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-9-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-22-9-2"  
class="td11"><!--l. 784--><p class="noindent" >Body  loads  are  supported.  Boundary  loads  are  not
supported now.                                                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-22-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-10-1"  
class="td11">Output                </td><td  style="white-space:wrap; text-align:left;" id="TBL-22-10-2"  
class="td11"><!--l. 793--><p class="noindent" >On output, the shell force (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">f</span></sub>), shell strain (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">s</span></sub>), shell momentum
(<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">m</span></sub>), and shell curvature (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">c</span></sub>) tensors in global coordinate
system are printed as vector form with 6 components, with the
following meaning:
<table 
class="align-star">
               <tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">f</span></sub></td>                <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>                  <td 
class="align-label"></td>             <td 
class="align-label">
             </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">s</span></sub></td>                 <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>                  <td 
class="align-label"></td>             <td 
class="align-label">
             </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">m</span></sub></td>               <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>             <td 
class="align-label"></td>             <td 
class="align-label">
             </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">c</span></sub></td>                 <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span></td>         <td 
class="align-label"></td>       <td 
class="align-label"></td></tr></table>
where        <span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">z</span></sub>        are        membrane        normal
deformations, <span 
class="cmmi-10">&gamma;</span><sub><span 
class="cmmi-7">zy</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">zx</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xy</span></sub> are (out of plane and in plane)
shear  componets,  <span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub>  are  curvatures,
<span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xy</span></sub>  are  integral  forces  (normal  and
shear forces), and <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub> are bending
moments. Please note, for example, that bending moment
<span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> is defined as <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> = <span 
class="cmex-10">&int;</span>
  <span 
class="cmmi-10">&sigma;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">z</span><span 
class="cmmi-10">&#x00A0;dz</span>, so it acts along the y-axis
and positive value causes tension in bottom layer.
</td></tr><tr  
 style="vertical-align:baseline;" id="TBL-22-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-11-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-22-11-2"  
class="td11"><!--l. 794--><p class="noindent" >0.                                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-22-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-22-12-2"  
class="td11"><!--l. 795--><p class="noindent" >Reliable                                                                  </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-22-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-13-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-22-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-22-14-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;21: </span><span  
class="content">cctplate3d element summary</span></div><!--tex4ht:label?: x11-2800121 -->
</div><hr class="endfloat" />
</div>
</div>
                                                                                           
                                                                                           
<!--l. 799--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.6.5   </span> <a 
 id="x11-290002.6.5"></a>RerShell Element</h5>
<!--l. 800--><p class="noindent" >Combination of CCT plate element (Mindlin hypothesis) with triangular plane stress element for membrane behavior.
The element curvature can be specified. Although element requires generally six DOFs per node, no stiffness to local
rotation along z-axis (rotation around element normal) is supplied. The element features are summarized in
Table&#x00A0;<a 
href="#x11-2900122">22<!--tex4ht:ref: rershellsummary --></a>.
<div class="center" 
>
<!--l. 806--><p class="noindent" >
<div class="table">
                                                                                           
                                                                                           
<!--l. 806--><p class="noindent" ><a 
 id="x11-2900122"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-23" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-23-1g"><col 
id="TBL-23-1"></colgroup><colgroup id="TBL-23-2g"><col 
id="TBL-23-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-23-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-23-1-2"  
class="td11"><!--l. 806--><p class="noindent" ><span 
class="cmbx-10">rershell</span>                                                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-23-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-23-2-2"  
class="td11"><!--l. 806--><p class="noindent" >Simple shell based on combination of CCT plate element
(Mindlin hypothesis) with triangular plane stress element.
element can be arbitrary positioned in space.                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-23-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-23-3-2"  
class="td11"><!--l. 806--><p class="noindent" >-                                                                            </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-23-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-23-4-2"  
class="td11"><!--l. 808--><p class="noindent" >Six dofs (u,v,w-displacements and u,v,w rotations) are in
general required in each node. Note, that although element
it requires generally six DOFs per node, no stiffness to local
rotation along z-axis (rotation around element normal) is
supplied.                                                                 </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-23-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-23-5-2"  
class="td11"><!--l. 809--><p class="noindent" >Linear      approximation      of      ratations,      quadratic
approximation of displacement.                                    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-23-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-23-6-2"  
class="td11"><!--l. 810--><p class="noindent" >Integration of all terms using one point formula.             </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-23-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-7-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-23-7-2"  
class="td11"><!--l. 811--><p class="noindent" >Layered cross section support.                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-23-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-8-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-23-8-2"  
class="td11"><!--l. 812--><p class="noindent" >Cross section thickness is required.                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-23-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-9-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-23-9-2"  
class="td11"><!--l. 813--><p class="noindent" >Body  loads  are  supported.  Boundary  loads  are  not
supported now.                                                         </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-23-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-10-1"  
class="td11">Output                </td><td  style="white-space:wrap; text-align:left;" id="TBL-23-10-2"  
class="td11"><!--l. 822--><p class="noindent" >On output, the shell force (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">f</span></sub>), shell strain (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">s</span></sub>), shell
momentum (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">m</span></sub>), and shell curvature (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">c</span></sub>) tensors in <span 
class="cmbx-10">global</span>
<span 
class="cmbx-10">coordinate system </span>are printed as vector form with 6
components, with the following meaning:
<table 
class="align-star">
               <tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">f</span></sub></td>                <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>                  <td 
class="align-label"></td>             <td 
class="align-label">
             </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">s</span></sub></td>                 <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>                  <td 
class="align-label"></td>             <td 
class="align-label">
             </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">m</span></sub></td>               <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>             <td 
class="align-label"></td>             <td 
class="align-label">
             </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">c</span></sub></td>                 <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span></td>         <td 
class="align-label"></td>       <td 
class="align-label"></td></tr></table>
where        <span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">z</span></sub>        are        membrane        normal
deformations, <span 
class="cmmi-10">&gamma;</span><sub><span 
class="cmmi-7">zy</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">zx</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xy</span></sub> are (out of plane and in plane)
shear  componets,  <span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub>  are  curvatures,
<span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xy</span></sub>  are  integral  forces  (normal  and
shear forces), and <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub> are bending
moments. Please note, for example, that bending moment
<span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> is defined as <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> = <span 
class="cmex-10">&int;</span>
  <span 
class="cmmi-10">&sigma;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">z</span><span 
class="cmmi-10">&#x00A0;dz</span>, so it acts along the y-axis
and positive value causes tension in bottom layer.
</td></tr><tr  
 style="vertical-align:baseline;" id="TBL-23-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-11-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-23-11-2"  
class="td11"><!--l. 823--><p class="noindent" >0.                                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-23-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-23-12-2"  
class="td11"><!--l. 824--><p class="noindent" >Reliable                                                                  </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-23-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-13-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-23-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-23-14-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;22: </span><span  
class="content">rershell element summary</span></div><!--tex4ht:label?: x11-2900122 -->
</div><hr class="endfloat" />
</div>
</div>
                                                                                           
                                                                                           
<!--l. 829--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.6.6   </span> <a 
 id="x11-300002.6.6"></a>tr_shell01 element</h5>
<!--l. 830--><p class="noindent" >Combination of CCT3D plate element (Mindlin hypothesis) with triangular plane stress element for membrane
behavior. It comes with complete set of 6 DOFs per node. The element features are summarized in Table&#x00A0;<a 
href="#x11-3000223">23<!--tex4ht:ref: trshell01summary --></a>.
<hr class="figure"><div class="figure" 
>
                                                                                           
                                                                                           
<a 
 id="x11-3000111"></a>
                                                                                           
                                                                                           
<!--l. 3--><p class="noindent" ><img 
src="elementlibmanual-figure10.png" alt="PIC"  
>
<br /> <div class="caption" 
><span class="id">Figure&#x00A0;11: </span><span  
class="content">Geometry of tr_shell01 element.</span></div><!--tex4ht:label?: x11-3000111 -->
                                                                                           
                                                                                           
<!--l. 839--><p class="indent" >   </div><hr class="endfigure">
<div class="center" 
>
<!--l. 841--><p class="noindent" >
<div class="table">
<!--l. 841--><p class="noindent" ><a 
 id="x11-3000223"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-24" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-24-1g"><col 
id="TBL-24-1"></colgroup><colgroup id="TBL-24-2g"><col 
id="TBL-24-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-24-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-24-1-2"  
class="td11"><!--l. 841--><p class="noindent" ><span 
class="cmbx-10">tr</span><span 
class="cmbx-10">_shell01</span>                                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-24-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-24-2-2"  
class="td11"><!--l. 841--><p class="noindent" >Triangular shell element combining CCT3D plate element
(Mindlin hypothesis) with triangular plane stress element
with rotational DOFs                                                </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-24-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-24-3-2"  
class="td11"><!--l. 841--><p class="noindent" >-                                                                            </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-24-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-24-4-2"  
class="td11"><!--l. 843--><p class="noindent" >Six dofs (u,v,w-displacements and u,v,w rotations) are in
general required in each node.                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-24-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-24-5-2"  
class="td11"><!--l. 844--><p class="noindent" >See description of cct and trplanstrrot elements              </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-24-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-24-6-2"  
class="td11"><!--l. 845--><p class="noindent" >Integration of all terms using one point formula.             </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-24-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-7-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-24-7-2"  
class="td11"><!--l. 846--><p class="noindent" >Layered cross section support.                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-24-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-8-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-24-8-2"  
class="td11"><!--l. 847--><p class="noindent" >Cross section thickness is required.                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-24-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-9-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-24-9-2"  
class="td11"><!--l. 848--><p class="noindent" >Body loads are supported. Boundary loads are supported
(only surface loads).                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-24-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-10-1"  
class="td11">Output                </td><td  style="white-space:wrap; text-align:left;" id="TBL-24-10-2"  
class="td11"><!--l. 857--><p class="noindent" >On output, the shell force (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">f</span></sub>), shell strain (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">s</span></sub>), shell
momentum (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">m</span></sub>), and shell curvature (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">c</span></sub>) tensors in <span 
class="cmbx-10">global</span>
<span 
class="cmbx-10">coordinate system </span>are printed as vector form with 6
components, with the following meaning:
<table 
class="align-star">
               <tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">f</span></sub></td>                <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>                  <td 
class="align-label"></td>             <td 
class="align-label">
             </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">s</span></sub></td>                 <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>                  <td 
class="align-label"></td>             <td 
class="align-label">
             </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">m</span></sub></td>               <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>             <td 
class="align-label"></td>             <td 
class="align-label">
             </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">c</span></sub></td>                 <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span></td>         <td 
class="align-label"></td>       <td 
class="align-label"></td></tr></table>
where        <span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">z</span></sub>        are        membrane        normal
deformations, <span 
class="cmmi-10">&gamma;</span><sub><span 
class="cmmi-7">zy</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">zx</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xy</span></sub> are (out of plane and in plane)
shear  componets,  <span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub>  are  curvatures,
<span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xy</span></sub>  are  integral  forces  (normal  and
shear forces), and <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub> are bending
moments. Please note, for example, that bending moment
<span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> is defined as <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> = <span 
class="cmex-10">&int;</span>
  <span 
class="cmmi-10">&sigma;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">z</span><span 
class="cmmi-10">&#x00A0;dz</span>, so it acts along the y-axis
and positive value causes tension in bottom layer.
</td></tr><tr  
 style="vertical-align:baseline;" id="TBL-24-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-11-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-24-11-2"  
class="td11"><!--l. 858--><p class="noindent" >0.                                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-24-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-24-12-2"  
class="td11"><!--l. 859--><p class="noindent" >Reliable                                                                  </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-24-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-13-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-24-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-24-14-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;23: </span><span  
class="content">tr_shell01 element summary</span></div><!--tex4ht:label?: x11-3000223 -->
</div><hr class="endfloat" />
</div>
                                                                                           
                                                                                           
</div>
   <h5 class="subsubsectionHead"><span class="titlemark">2.6.7   </span> <a 
 id="x11-310002.6.7"></a>tr_shell02 element</h5>
<!--l. 864--><p class="noindent" >Combination of thin-plate DKT plate element with plane stress element (TrPlanestressRotAllman).
This element comes with complete set of 6 DOFs per node. The element features are summarized in
Table&#x00A0;<a 
href="#x11-3100124">24<!--tex4ht:ref: trshell02summary --></a>.
<div class="center" 
>
<!--l. 867--><p class="noindent" >
<div class="table">
                                                                                           
                                                                                           
<!--l. 867--><p class="noindent" ><a 
 id="x11-3100124"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-25" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-25-1g"><col 
id="TBL-25-1"></colgroup><colgroup id="TBL-25-2g"><col 
id="TBL-25-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-25-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-25-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-25-1-2"  
class="td11"><!--l. 867--><p class="noindent" ><span 
class="cmbx-10">tr</span><span 
class="cmbx-10">_shell02</span>                                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-25-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-25-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-25-2-2"  
class="td11"><!--l. 867--><p class="noindent" >Triangular  shell  element  combining  DKT  plate  element
with triangular plane stress element with rotational DOFs</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-25-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-25-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-25-3-2"  
class="td11"><!--l. 867--><p class="noindent" >-                                                                            </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-25-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-25-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-25-4-2"  
class="td11"><!--l. 869--><p class="noindent" >Six dofs (u,v,w-displacements and u,v,w rotations) are in
general required in each node.                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-25-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-25-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-25-5-2"  
class="td11"><!--l. 870--><p class="noindent" >See description of cct and trplanstrrot elements              </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-25-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-25-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-25-6-2"  
class="td11"><!--l. 871--><p class="noindent" >4  integration  points  necessary,  use  &#8221;NIP  4&#8221;  in  element
record.                                                                    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-25-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-25-7-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-25-7-2"  
class="td11"><!--l. 873--><p class="noindent" >Cross section thickness is required.                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-25-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-25-8-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-25-8-2"  
class="td11"><!--l. 874--><p class="noindent" >Body loads are supported. Boundary loads are supported
(only surface loads).                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-25-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-25-9-1"  
class="td11">Output                </td><td  style="white-space:wrap; text-align:left;" id="TBL-25-9-2"  
class="td11"><!--l. 883--><p class="noindent" >On output, the shell force (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">f</span></sub>), shell strain (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">s</span></sub>), shell
momentum (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">m</span></sub>), and shell curvature (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">c</span></sub>) tensors in <span 
class="cmbx-10">global</span>
<span 
class="cmbx-10">coordinate system </span>are printed as vector form with 6
components, with the following meaning:
<table 
class="align-star">
               <tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">f</span></sub></td>                <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>                  <td 
class="align-label"></td>             <td 
class="align-label">
             </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">s</span></sub></td>                 <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>                  <td 
class="align-label"></td>             <td 
class="align-label">
             </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">m</span></sub></td>               <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>             <td 
class="align-label"></td>             <td 
class="align-label">
             </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">c</span></sub></td>                 <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span></td>         <td 
class="align-label"></td>       <td 
class="align-label"></td></tr></table>
where        <span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">z</span></sub>        are        membrane        normal
deformations, <span 
class="cmmi-10">&gamma;</span><sub><span 
class="cmmi-7">zy</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">zx</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xy</span></sub> are (out of plane and in plane)
shear  componets,  <span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub>  are  curvatures,
<span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xy</span></sub>  are  integral  forces  (normal  and
shear forces), and <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub> are bending
moments. Please note, for example, that bending moment
<span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> is defined as <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> = <span 
class="cmex-10">&int;</span>
  <span 
class="cmmi-10">&sigma;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">z</span><span 
class="cmmi-10">&#x00A0;dz</span>, so it acts along the y-axis
and positive value causes tension in bottom layer.
</td></tr><tr  
 style="vertical-align:baseline;" id="TBL-25-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-25-10-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-25-10-2"  
class="td11"><!--l. 884--><p class="noindent" >0.                                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-25-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-25-11-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-25-11-2"  
class="td11"><!--l. 885--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-25-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-25-12-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-25-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-25-13-1"  
class="td11">                </td></tr></table></div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;24: </span><span  
class="content">tr_shell02 element summary</span></div><!--tex4ht:label?: x11-3100124 -->
</div><hr class="endfloat" />
</div>
</div>
<!--l. 890--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.6.8   </span> <a 
 id="x11-320002.6.8"></a>Quad1Mindlin Element</h5>
<!--l. 891--><p class="noindent" >This class implements an quadrilateral, bilinear, four-node Mindlin plate. This type of element exhibit strong shear
                                                                                           
                                                                                           
locking (thin plates exhibit almost no bending). Implements the lumped mass matrix. The element features are
summarized in Table&#x00A0;<a 
href="#x11-3200125">25<!--tex4ht:ref: quad1mindlinsummary --></a>.
<div class="center" 
>
<!--l. 895--><p class="noindent" >
<div class="table">
<!--l. 895--><p class="noindent" ><a 
 id="x11-3200125"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-26" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-26-1g"><col 
id="TBL-26-1"></colgroup><colgroup id="TBL-26-2g"><col 
id="TBL-26-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-26-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-26-1-2"  
class="td11"><!--l. 895--><p class="noindent" ><span 
class="cmbx-10">quad1mindlin</span>                                                 </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-26-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-26-2-2"  
class="td11"><!--l. 895--><p class="noindent" >Quadrilateral, bilinear, four-node Mindlin plate               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-26-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-26-3-2"  
class="td11"><!--l. 895--><p class="noindent" >[<span 
class="cmtt-10">NIP</span>&#x00A0;#<span 
class="cmr-5">(in)</span>]                                                               </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-26-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-26-4-2"  
class="td11"><!--l. 897--><p class="noindent" >Three dofs (w-displacement, u and v - rotation) are required
in each node.                                                           </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-26-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-26-5-2"  
class="td11"><!--l. 898--><p class="noindent" >Linear for all unknowns.                                             </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-26-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-26-6-2"  
class="td11"><!--l. 899--><p class="noindent" >Default uses 4 integration points. No reduced integration is
used, as it causes numerical problems.                           </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-26-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-7-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-26-7-2"  
class="td11"><!--l. 900--><p class="noindent" >Layered cross section support.                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-26-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-8-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-26-8-2"  
class="td11"><!--l. 901--><p class="noindent" >Cross section thickness is required.                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-26-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-9-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-26-9-2"  
class="td11"><!--l. 902--><p class="noindent" >Dead weight loads, and edge loads are supported.            </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-26-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-10-1"  
class="td11">Output                </td><td  style="white-space:wrap; text-align:left;" id="TBL-26-10-2"  
class="td11"><!--l. 909--><p class="noindent" >On output, the generalized shell strain/force momentum
vectors in global coordinate system are printed, with the
following meaning:
<table 
class="align-star">
          <tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">&#x03B5;</span></sub></td>          <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>               <td 
class="align-label"></td>        <td 
class="align-label">
        </td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">&sigma;</span></sub></td>          <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmsy-10">}</span></td>    <td 
class="align-label"></td>    <td 
class="align-label"></td></tr></table>
where                                                         <span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">xy</span></sub>
are membrane in plane normal deformations, <span 
class="cmmi-10">&gamma;</span><sub><span 
class="cmmi-7">zx</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xz</span></sub> are
(out of plane and in plane) shear componets, <span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub> are
curvatures, <span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,q</span><sub><span 
class="cmmi-7">yz</span></sub> are integral forces (normal
and shear forces), and <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub> are bending moments.
Please  note,  for  example,  that  bending  moment  <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub>  is
defined as <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> = <span 
class="cmex-10">&int;</span>
  <span 
class="cmmi-10">&sigma;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">z</span><span 
class="cmmi-10">&#x00A0;dz</span>, so it acts along the y-axis and
positive value causes tension in bottom layer.
</td></tr><tr  
 style="vertical-align:baseline;" id="TBL-26-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-11-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-26-11-2"  
class="td11"><!--l. 910--><p class="noindent" >0.                                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-26-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-12-1"  
class="td11">Reference             </td><td  style="white-space:wrap; text-align:left;" id="TBL-26-12-2"  
class="td11"><!--l. 911--><p class="noindent" ><span class="cite">[<a 
href="elementlibmanualli3.html#XRobertCook1989">1</a>]</span>                                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-26-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-13-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-26-13-2"  
class="td11"><!--l. 912--><p class="noindent" >Experimental                                                           </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-26-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-14-1"  
class="td11">                </td></tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-26-15-"><td  style="white-space:nowrap; text-align:left;" id="TBL-26-15-1"  
class="td11"></td></tr></table>
</div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;25: </span><span  
class="content">quad1mindlin element summary</span></div><!--tex4ht:label?: x11-3200125 -->
</div><hr class="endfloat" />
</div>
</div>
                                                                                           
                                                                                           
<!--l. 919--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.6.9   </span> <a 
 id="x11-330002.6.9"></a>Tr2Shell7 Element</h5>
<!--l. 920--><p class="noindent" >This class implements a triangular, quadratic, six-node shell element. The element is a so-called seven parameter shell
with seven dofs per node &#8211; a displacement field (3 dofs), an extensible director field (3 dofs) and a seventh dof
representing inhomogenous thickness strain. This last parameter is included in the model in order to deal with
volumetric/Poisson lock effects.
<!--l. 922--><p class="indent" >   The element features are summarized in Table&#x00A0;<a 
href="#x11-3200125">25<!--tex4ht:ref: quad1mindlinsummary --></a>.
<div class="center" 
>
<!--l. 924--><p class="noindent" >
<div class="table">
<!--l. 924--><p class="noindent" ><a 
 id="x11-3300126"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-27" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-27-1g"><col 
id="TBL-27-1"></colgroup><colgroup id="TBL-27-2g"><col 
id="TBL-27-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-27-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-27-1-2"  
class="td11"><!--l. 924--><p class="noindent" ><span 
class="cmbx-10">tr2shell7</span>                                                        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-27-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-27-2-2"  
class="td11"><!--l. 924--><p class="noindent" >Triangular, quadratic, six-node shell with 7 dofs/node      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-27-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-27-3-2"  
class="td11"><!--l. 924--><p class="noindent" >[<span 
class="cmtt-10">NIP</span>&#x00A0;#<span 
class="cmr-5">(in)</span>]                                                               </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-27-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-27-4-2"  
class="td11"><!--l. 926--><p class="noindent" >Seven dofs (displacement in u, v and w-direction; change
in director field in u, v and w-direction; and inhomgenous
thickness stretch) are required in each node.                   </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-27-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-27-5-2"  
class="td11"><!--l. 927--><p class="noindent" >Quadratic for all unknowns.                                        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-27-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-27-6-2"  
class="td11"><!--l. 928--><p class="noindent" >Default uses 6 integration points in the midsurface plane.
Number of integration points in the thickness direction is
determined by the Layered cross section.                       </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-27-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-7-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-27-7-2"  
class="td11"><!--l. 929--><p class="noindent" >Layered cross section support.                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-27-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-8-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-27-8-2"  
class="td11"><!--l. 930--><p class="noindent" >This element must be used with a Layered cross section.   </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-27-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-9-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-27-9-2"  
class="td11"><!--l. 931--><p class="noindent" >Edge loads, constant pressure loads and surface loads are
supported.                                                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-27-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-10-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-27-10-2"  
class="td11"><!--l. 932--><p class="noindent" >Not        applicable.        The        implementation        is
for large defomrations and hence geometrical nonlinearities
will always be present, regardless the value of Nlgeo.        </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-27-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-11-1"  
class="td11">Reference             </td><td  style="white-space:wrap; text-align:left;" id="TBL-27-11-2"  
class="td11"><!--l. 933--><p class="noindent" ><span class="cite">[<a 
href="elementlibmanualli3.html#XRagnarLarsson2011">3</a>]</span>                                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-27-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-12-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-27-12-2"  
class="td11"><!--l. 934--><p class="noindent" >Experimental                                                           </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-27-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-13-1"  
class="td11">                </td></tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-27-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-27-14-1"  
class="td11"></td></tr></table>
</div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;26: </span><span  
class="content">tr2shell7 element summary</span></div><!--tex4ht:label?: x11-3300126 -->
</div><hr class="endfloat" />
</div>
</div>
<!--l. 939--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.6.10   </span> <a 
 id="x11-340002.6.10"></a>MITC4Shell Element</h5>
<!--l. 940--><p class="noindent" >A four-node quadrilateral shell element formulated using three-dimensional continuum mechanics theory degenerated
to shell behaviour. The element is applicable to thick and thin shells as the &#8220;mixed interpolation of tensorial
components&#8221; (MITC) approach is used to remove shear locking. The implementation is based on the following paper:
Dvorkin, E.N., Bathe, K.J.: A continuum mechanics based four-node shell element for general non-linear analysis,
Eng.Comput., Vol.1, 77-88, 1984.
<!--l. 942--><p class="indent" >   Although element requires generally six DOFs per node, no stiffness to local rotation along z-axis (rotation around
director vector) is supplied. The element features are summarized in Table&#x00A0;<a 
href="#x11-3400127">27<!--tex4ht:ref: mitc4shellsummary --></a>.
<div class="center" 
>
<!--l. 944--><p class="noindent" >
<div class="table">
                                                                                           
                                                                                           
<!--l. 944--><p class="noindent" ><a 
 id="x11-3400127"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-28" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-28-1g"><col 
id="TBL-28-1"></colgroup><colgroup id="TBL-28-2g"><col 
id="TBL-28-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-28-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-28-1-2"  
class="td11"><!--l. 944--><p class="noindent" ><span 
class="cmbx-10">mitc4shell</span>                                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-28-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-28-2-2"  
class="td11"><!--l. 944--><p class="noindent" >Quadrilateral, bilinear, four-node shell element using the
MITC technique.                                                      </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-28-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-28-3-2"  
class="td11"><!--l. 944--><p class="noindent" >[<span 
class="cmtt-10">NIP</span>&#x00A0;#<span 
class="cmr-5">(in)</span>] [<span 
class="cmtt-10">NIPZ</span>&#x00A0;#<span 
class="cmr-5">(in)</span>] [<span 
class="cmtt-10">directorType</span>&#x00A0;#<span 
class="cmr-5">(in)</span>]                  </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-28-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-4-1"  
class="td11">Parameters           </td><td  style="white-space:wrap; text-align:left;" id="TBL-28-4-2"  
class="td11"><!--l. 945--><p class="noindent" ><span 
class="cmtt-10">NIP</span>: allows to set the number of integration points in local
x-y plane (default 4).                                                 </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-28-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-5-1"  
class="td11">                </td><td  style="white-space:wrap; text-align:left;" id="TBL-28-5-2"  
class="td11"><!--l. 946--><p class="noindent" ><span 
class="cmtt-10">NIPZ</span>: allows to set the number of integration points in local
z-direction (default 2).                                               </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-28-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-6-1"  
class="td11">                </td><td  style="white-space:wrap; text-align:left;" id="TBL-28-6-2"  
class="td11"><!--l. 947--><p class="noindent" ><span 
class="cmtt-10">directorType</span>:  allows  to  set  director  vectors.  Director
vectors can be set as normal to the plane (<span 
class="cmtt-10">directorType</span>
= 0, default), or calculated for each node as an average of
neighbouring elements of same crosssection (<span 
class="cmtt-10">directorType</span>
= 1), or can be specified at crosssection (<span 
class="cmtt-10">directorType</span>
=2).                                                                       </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-28-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-7-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-28-7-2"  
class="td11"><!--l. 948--><p class="noindent" >Six dofs (u,v,w-displacements and u,v,w rotations) are in
general required in each node. Note, that although element
requires generally six DOFs per node, no stiffness to local
rotation along z-axis (rotation around director vector) is
supplied.                                                                 </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-28-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-8-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-28-8-2"  
class="td11"><!--l. 949--><p class="noindent" >Linear approximation of displacements and rotations.       </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-28-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-9-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-28-9-2"  
class="td11"><!--l. 950--><p class="noindent" >Integration of all terms using Gauss integration formula in 8
points (default) or specified using <span 
class="cmtt-10">NIP </span>and <span 
class="cmtt-10">NIPZ </span>parameters.</td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-28-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-10-1"  
class="td11">Features              </td><td  style="white-space:wrap; text-align:left;" id="TBL-28-10-2"  
class="td11"><!--l. 951--><p class="noindent" >Variable cross section support.                                     </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-28-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-11-1"  
class="td11">CS properties       </td><td  style="white-space:wrap; text-align:left;" id="TBL-28-11-2"  
class="td11"><!--l. 952--><p class="noindent" >Cross section thickness is required (measured along director
vector).  Director  vectors  components  may  be  specified
[<span 
class="cmtt-10">directorx</span>&#x00A0;#<span 
class="cmr-5">(in)</span>][<span 
class="cmtt-10">directory</span>&#x00A0;#<span 
class="cmr-5">(in)</span>][<span 
class="cmtt-10">directorz</span>&#x00A0;#<span 
class="cmr-5">(in)</span>]    in
case of <span 
class="cmtt-10">directorType 2</span>.                                            </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-28-12-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-12-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-28-12-2"  
class="td11"><!--l. 953--><p class="noindent" >Body and boundary loads are supported.                       </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-28-13-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-13-1"  
class="td11">Output                </td><td  style="white-space:wrap; text-align:left;" id="TBL-28-13-2"  
class="td11"><!--l. 966--><p class="noindent" >On output, the shell force (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">f</span></sub>), shell momentum (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">m</span></sub>), shell
strain (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">s</span></sub>), shell curvature (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">c</span></sub>), strain (<span 
class="cmmi-10">&#x03B5;</span>), and stress
(<span 
class="cmmi-10">&sigma;</span>) tensors in <span 
class="cmbx-10">global coordinate system </span>are printed
as vector form with 6 components, with the following
meaning:
<table 
class="align-star">
<tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">f</span></sub></td> <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>     <td 
class="align-odd"></td>                           <td 
class="align-even"></td><td 
class="align-label"></td><td 
class="align-label">
</td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">m</span></sub></td><td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td><td 
class="align-odd"></td>                           <td 
class="align-even"></td><td 
class="align-label"></td><td 
class="align-label">
</td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">s</span></sub></td> <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td>     <td 
class="align-odd"></td>                           <td 
class="align-even"></td><td 
class="align-label"></td><td 
class="align-label">
</td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">c</span></sub></td> <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span></td>   <td 
class="align-odd"></td>              <td 
class="align-even"></td><td 
class="align-label"></td><td 
class="align-label">
</td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">&#x03B5;</span></td>  <td 
class="align-even"> =</td>                                        <td 
class="align-odd"><span 
class="cmsy-10">{</span><span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">zx</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">,</span></td><td 
class="align-even"></td><td 
class="align-label"></td><td 
class="align-label"></td><td 
class="align-label">
</td></tr><tr><td 
class="align-odd"><span 
class="cmmi-10">&sigma;</span></td>  <td 
class="align-even"> = <span 
class="cmsy-10">{</span><span 
class="cmmi-10">&sigma;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&sigma;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&sigma;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&sigma;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&sigma;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&sigma;</span><sub><span 
class="cmmi-7">xy</span></sub><span 
class="cmsy-10">}</span><span 
class="cmmi-10">.</span></td>    <td 
class="align-odd"></td>                           <td 
class="align-even"></td><td 
class="align-label"></td><td 
class="align-label"> </td></tr> </table>
where  <span 
class="cmmi-10">n</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,n</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,v</span><sub><span 
class="cmmi-7">xy</span></sub>  are  integral  forces  (normal
and  shear  forces),  and  <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,m</span><sub><span 
class="cmmi-7">xy</span></sub>  are
bending   moments,   <span 
class="cmmi-10">&#x03B5;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&#x03B5;</span><sub><span 
class="cmmi-7">z</span></sub>   are   membrane   normal
deformations, <span 
class="cmmi-10">&gamma;</span><sub><span 
class="cmmi-7">zy</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">zx</span></sub><span 
class="cmmi-10">,&gamma;</span><sub><span 
class="cmmi-7">xy</span></sub> are (out of plane and in plane)
shear  componets,  <span 
class="cmmi-10">&kappa;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">y</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">z</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">yz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xz</span></sub><span 
class="cmmi-10">,&kappa;</span><sub><span 
class="cmmi-7">xy</span></sub>  are  curvatures.
Please note, for example, the bending moment <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> is defined
as <span 
class="cmmi-10">m</span><sub><span 
class="cmmi-7">x</span></sub> = <span 
class="cmex-10">&int;</span>
  <span 
class="cmmi-10">&sigma;</span><sub><span 
class="cmmi-7">x</span></sub><span 
class="cmmi-10">z</span><span 
class="cmmi-10">&#x00A0;dz</span>, so it acts along the y-axis and positive
value causes tension in bottom layer (positive z-coordinate).
The  shell  force  (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">f</span></sub>),  shell  momentum  (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">m</span></sub>),  shell  strain
(<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">s</span></sub>),  and  shell  curvature  (<span 
class="cmmi-10">s</span><sub><span 
class="cmmi-7">c</span></sub>)  tensors  are  evaluated  at
the midplane of the element (thus are constant along the
thickness) while the strain (<span 
class="cmmi-10">&#x03B5;</span>), and stress (<span 
class="cmmi-10">&sigma;</span>) tensors are
evaluated at each Gausspoint.
</td></tr><tr  
 style="vertical-align:baseline;" id="TBL-28-14-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-14-1"  
class="td11">Nlgeo                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-28-14-2"  
class="td11"><!--l. 967--><p class="noindent" >0.                                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-28-15-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-15-1"  
class="td11">Status                 </td><td  style="white-space:wrap; text-align:left;" id="TBL-28-15-2"  
class="td11"><!--l. 968--><p class="noindent" >-                                                                            </td></tr><tr  
 style="vertical-align:baseline;" id="TBL-28-16-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-16-1"  
class="td11"> </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-28-17-"><td  style="white-space:nowrap; text-align:left;" id="TBL-28-17-1"  
class="td11">                </td></tr></table>                                                                        
                                                                                           
</div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;27: </span><span  
class="content">mitc4shell element summary</span></div><!--tex4ht:label?: x11-3400127 -->
</div><hr class="endfloat" />
</div>
</div>
<!--l. 972--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.6.11   </span> <a 
 id="x11-350002.6.11"></a>Sub-soil Elements</h5>
<!--l. 973--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.6.12   </span> <a 
 id="x11-360002.6.12"></a>quad1plateSubsoil Element</h5>
<!--l. 974--><p class="noindent" >This class implements an quadrilateral, bilinear, four-node plate subsoil element. Typically this element is combined
with suitable plate element with quadrilateral geometry to model plate element on (elastic) subsoill foundation, but it
can be used alone. The element geometry should be define in xy plane. The element features are summarized in
Table&#x00A0;<a 
href="#x11-3600128">28<!--tex4ht:ref: quad1platesubsoilsummary --></a>.
<div class="center" 
>
<!--l. 979--><p class="noindent" >
<div class="table">
<!--l. 979--><p class="noindent" ><a 
 id="x11-3600128"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-29" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-29-1g"><col 
id="TBL-29-1"></colgroup><colgroup id="TBL-29-2g"><col 
id="TBL-29-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-29-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-29-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-29-1-2"  
class="td11"><!--l. 979--><p class="noindent" ><span 
class="cmbx-10">quad1plateSubsoil</span>                                           </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-29-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-29-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-29-2-2"  
class="td11"><!--l. 979--><p class="noindent" >Quadrilateral, bilinear, four-node sub-soil plate element    </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-29-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-29-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-29-3-2"  
class="td11"><!--l. 979--><p class="noindent" >                                                 </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-29-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-29-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-29-4-2"  
class="td11"><!--l. 981--><p class="noindent" >One dof (w-displacement) is required in each node.          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-29-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-29-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-29-5-2"  
class="td11"><!--l. 982--><p class="noindent" >Linear for transwersal displacement.                             </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-29-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-29-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-29-6-2"  
class="td11"><!--l. 983--><p class="noindent" >4 integration points.                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-29-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-29-7-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-29-7-2"  
class="td11"><!--l. 984--><p class="noindent" >Surface load support.                                                 </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-29-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-29-8-1"  
class="td11">Note                   </td><td  style="white-space:wrap; text-align:left;" id="TBL-29-8-2"  
class="td11"><!--l. 985--><p class="noindent" >Requires   material   model   with   2dPlateSubSoil   mode
support.                                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-29-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-29-9-1"  
class="td11">Reference             </td><td  style="white-space:wrap; text-align:left;" id="TBL-29-9-2"  
class="td11"><!--l. 986--><p class="noindent" ><span class="cite">[<a 
href="elementlibmanualli3.html#XBittnarSejnoha1996">2</a>]</span>                                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-29-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-29-10-1"  
class="td11">                </td></tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-29-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-29-11-1"  
class="td11"></td></tr></table>
</div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;28: </span><span  
class="content">quad1platesubsoil element summary</span></div><!--tex4ht:label?: x11-3600128 -->
</div><hr class="endfloat" />
</div>
</div>
<!--l. 989--><p class="noindent" >
   <h5 class="subsubsectionHead"><span class="titlemark">2.6.13   </span> <a 
 id="x11-370002.6.13"></a>Tria1PlateSubSoil Element</h5>
<!--l. 990--><p class="noindent" >This class implements an quadrilateral, bilinear, four-node plate subsoil element. Typically this element is combined
with suitable plate element with quadrilateral geometry to model plate element on (elastic) subsoill foundation, but it
can be used alone. The element geometry should be define in xy plane. The element features are summarized in
Table&#x00A0;<a 
href="#x11-3600128">28<!--tex4ht:ref: quad1platesubsoilsummary --></a>.
<div class="center" 
>
<!--l. 995--><p class="noindent" >
<div class="table">
                                                                                           
                                                                                           
<!--l. 995--><p class="noindent" ><a 
 id="x11-3700129"></a><hr class="float"><div class="float" 
> <!--tex4ht:inline--><div class="tabular"> <table id="TBL-30" class="tabular" 
cellspacing="0" cellpadding="0" rules="groups" 
><colgroup id="TBL-30-1g"><col 
id="TBL-30-1"></colgroup><colgroup id="TBL-30-2g"><col 
id="TBL-30-2"></colgroup><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-30-1-"><td  style="white-space:nowrap; text-align:left;" id="TBL-30-1-1"  
class="td11"><span 
class="cmbx-10">Keyword         </span></td><td  style="white-space:wrap; text-align:left;" id="TBL-30-1-2"  
class="td11"><!--l. 995--><p class="noindent" ><span 
class="cmbx-10">tria1platesubsoil</span>                                             </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-30-2-"><td  style="white-space:nowrap; text-align:left;" id="TBL-30-2-1"  
class="td11">Description          </td><td  style="white-space:wrap; text-align:left;" id="TBL-30-2-2"  
class="td11"><!--l. 995--><p class="noindent" >Tringular,  three-node  sub-soil  plate  element  with  linear
interpolation                                                            </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-30-3-"><td  style="white-space:nowrap; text-align:left;" id="TBL-30-3-1"  
class="td11">Specific parameters</td><td  style="white-space:wrap; text-align:left;" id="TBL-30-3-2"  
class="td11"><!--l. 995--><p class="noindent" >                                                 </td>
</tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-30-4-"><td  style="white-space:nowrap; text-align:left;" id="TBL-30-4-1"  
class="td11">Unknowns            </td><td  style="white-space:wrap; text-align:left;" id="TBL-30-4-2"  
class="td11"><!--l. 996--><p class="noindent" >One dof (w-displacement) is required in each node.          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-30-5-"><td  style="white-space:nowrap; text-align:left;" id="TBL-30-5-1"  
class="td11">Approximation      </td><td  style="white-space:wrap; text-align:left;" id="TBL-30-5-2"  
class="td11"><!--l. 997--><p class="noindent" >Linear for transwersal displacement.                             </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-30-6-"><td  style="white-space:nowrap; text-align:left;" id="TBL-30-6-1"  
class="td11">Integration           </td><td  style="white-space:wrap; text-align:left;" id="TBL-30-6-2"  
class="td11"><!--l. 998--><p class="noindent" >1 integration points.                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-30-7-"><td  style="white-space:nowrap; text-align:left;" id="TBL-30-7-1"  
class="td11">Loads                  </td><td  style="white-space:wrap; text-align:left;" id="TBL-30-7-2"  
class="td11"><!--l. 999--><p class="noindent" >Surface load support.                                                 </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-30-8-"><td  style="white-space:nowrap; text-align:left;" id="TBL-30-8-1"  
class="td11">Note                   </td><td  style="white-space:wrap; text-align:left;" id="TBL-30-8-2"  
class="td11"><!--l. 1000--><p class="noindent" >Requires   material   model   with   2dPlateSubSoil   mode
support.                                                                  </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-30-9-"><td  style="white-space:nowrap; text-align:left;" id="TBL-30-9-1"  
class="td11">Reference             </td><td  style="white-space:wrap; text-align:left;" id="TBL-30-9-2"  
class="td11"><!--l. 1001--><p class="noindent" ><span class="cite">[<a 
href="elementlibmanualli3.html#XBittnarSejnoha1996">2</a>]</span>                                                                          </td>
</tr><tr  
 style="vertical-align:baseline;" id="TBL-30-10-"><td  style="white-space:nowrap; text-align:left;" id="TBL-30-10-1"  
class="td11">                </td></tr><tr 
class="hline"><td><hr></td><td><hr></td></tr><tr  
 style="vertical-align:baseline;" id="TBL-30-11-"><td  style="white-space:nowrap; text-align:left;" id="TBL-30-11-1"  
class="td11"></td></tr></table>
</div>
<br /> <div class="caption" 
><span class="id">Table&#x00A0;29: </span><span  
class="content">tria1platesubsoil element summary</span></div><!--tex4ht:label?: x11-3700129 -->
</div><hr class="endfloat" />
</div>
</div>
                                                                                           
                                                                                           
                                                                                           
                                                                                           
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